$(30$ points$)$ Problem $1$: Recursion.

$($a$)(10$ points$)$ Describe a linear recursive algorithm $($using pseudocode$)$ for finding the number of vowels in a string $S($the vowels are a$,$e$,,i,o,u).$ What is your running time and space usage?

$($b$)(10$ points$)$ Recall that in class we solved the problem of computing powers using a recursive algorithm. Draw the recursion tree for the computation Power $(2,18).$

$($c$)(10$ points$)$ Write a java program that uses a recursive method, mult$(),$ that performs multiplication of $x$ and $y$ by adding $x$ to itself $y$ times. Its arguments are $x$ and $y$ and its return value is the product of $x$ and $y.$ Write such a method and a main$()$ program to call it$.$